scholarly journals Analytical Solutions of the Supersaturation Equation for a Warm Cloud

2016 ◽  
Vol 73 (9) ◽  
pp. 3453-3465 ◽  
Author(s):  
B. J. Devenish ◽  
K. Furtado ◽  
D. J. Thomson
Keyword(s):  
Vertical Velocity ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Droplet Radius ◽  
Scaling Relation ◽  
Number Density ◽  
Initial Droplet ◽  
Warm Cloud ◽  
Small Times ◽  
Analytical Expressions

Abstract Approximate solutions of the supersaturation equation are derived for a warm cloud. These solutions take account of the growth of the droplet radius but are only valid for small times. However, the validity of the solutions extends far enough to obtain reasonable estimates of the maximum supersaturation smax. It is shown that when the initial droplet radius and supersaturation are sufficiently small the scaling relation smax ∝ w3/4N−1/2 is obtained, where w is the vertical velocity and N is the droplet number density, in agreement with previous results. The range of validity of this result is discussed and other analytical expressions are derived when this result is not valid. It is shown that these analytical expressions generally agree well with numerical solutions.

The variability of warm cloud droplet radius induced by aerosols and water vapor in Shanghai from MODIS observations

2021 ◽  
Vol 253 ◽  
pp. 105470
Author(s):  
Qiong Liu ◽  
Shengyang Duan ◽  
Qianshan He ◽  
Yonghang Chen ◽  
Hua Zhang ◽  
...  
Keyword(s):  
Water Vapor ◽  
Cloud Droplet ◽  
Droplet Radius ◽  
Warm Cloud

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method

2013 ◽  
Vol 430 ◽  
pp. 22-26 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu ◽  
Traian Marinca
Keyword(s):  
Small Parameter ◽  
Cantilever Beam ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Lumped Mass ◽  
Engineering Problems ◽  
Approximate Frequency ◽  
Optimal Homotopy Asymptotic Method ◽  
Analytical Expressions

The response of a cantilever beam with a lumped mass attached to its free end subject to harmonical excitation at the base is investigated by means of the Optimal Homotopy Asymptotic Method (OHAM). Approximate accurate analytical expressions for the solutions and for approximate frequency are determined. This method does not require any small parameter in the equation. The obtained results prove that our method is very accurate, effective and simple for investigation of such engineering problems.

scholarly journals Sensitivities of Lagrangian modelling of mid-latitude cirrus clouds to trajectory data quality

2015 ◽  
Vol 15 (13) ◽  
pp. 7429-7447 ◽  
Author(s):  
E. Kienast-Sjögren ◽  
A. K. Miltenberger ◽  
B. P. Luo ◽  
T. Peter
Keyword(s):  
Cooling Rate ◽  
Temporal Resolution ◽  
Vertical Velocity ◽  
Input Data ◽  
Temperature Fluctuations ◽  
Box Model ◽  
Specific Humidity ◽  
Number Density ◽  
Ice Nuclei ◽  
Nuclei Number

Abstract. Simulations of cirrus are subject to uncertainties in model physics and meteorological input data. Here we model cirrus clouds along air mass trajectories, whose extinction has been measured with an elastic backscatter lidar at Jungfraujoch research station in the Swiss Alps, with a microphysical stacked box model. The sensitivities of these simulations to input data uncertainties (trajectory resolution, unresolved vertical velocities, ice nuclei number density and upstream specific humidity) are investigated. Variations in the temporal resolution of the wind field data (COSMO-Model at 2.2 km resolution) between 20 s and 1 h have only a marginal impact on the trajectory path, while the representation of the vertical velocity variability and therefore the cooling rate distribution are significantly affected. A temporal resolution better than 5 min must be chosen in order to resolve cooling rates required to explain the measured extinction. A further increase in the temporal resolution improves the simulation results slightly. The close match between the modelled and observed extinction profile for high-resolution trajectories suggests that the cooling rate spectra calculated by the COSMO-2 model suffice on the selected day. The modelled cooling rate spectra are, however, characterized by significantly lower vertical velocity amplitudes than those found previously in some aircraft campaigns (SUCCESS, MACPEX). A climatological analysis of the vertical velocity amplitude in the Alpine region based on COSMO-2 analyses and balloon sounding data suggests large day-to-day variability in small-scale temperature fluctuations. This demonstrates the necessity to apply numerical weather prediction models with high spatial and temporal resolutions in cirrus modelling, whereas using climatological means for the amplitude of the unresolved air motions does generally not suffice. The box model simulations further suggest that uncertainties in the upstream specific humidity (± 10 % of the model prediction) and in the ice nuclei number density (0–100 L−1) are more important for the modelled cirrus cloud than the unresolved temperature fluctuations if temporally highly resolved trajectories are used. For the presented case the simulations are incompatible with ice nuclei number densities larger than 20 L−1 and insensitive to variations below this value.

scholarly journals Semi-Analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian Expansion of Distance Operators W= RC1-nRD1-M, RC1-Nr12-M, R12-Nr13-M

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Gaussian Function ◽  
Approximate Solutions ◽  
Equation System ◽  
Least Square ◽  
Position Vector ◽  
Least Square Fit ◽  
Analytic Expressions ◽  
The One ◽  
Analytical Expressions ◽  
Coulomb Integrals

The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)…rho(rk) W(r1,…,rk) dr1…drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases – mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u≠1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrödinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

An Elongating String Under the Action of a Transverse Force

1957 ◽  
Vol 24 (4) ◽  
pp. 609-616
Author(s):  
Werner Goldsmith
Keyword(s):  
Method Of Characteristics ◽  
Numerical Solutions ◽  
Transverse Force ◽  
Transverse Displacement ◽  
Fixed Frequency ◽  
Constant Acceleration ◽  
Forcing Function ◽  
Subsonic Regime ◽  
Arbitrary Amplitude ◽  
Analytical Expressions

Abstract The motion of a uniform undamped flexible string whose length increases with time has been investigated when an arbitrary time-dependent force acts transversely at the free end. The method of characteristics has been employed to derive analytical expressions for the transverse displacement in the subsonic regime. Cases are considered when the free end of the wire moves either at constant velocity or at constant acceleration. Numerical solutions are presented in dimensionless form for a sinusoidal forcing function of arbitrary amplitude and fixed frequency. The possibility of the existence of resonances in the string has been examined.

Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment

2011 ◽  
Vol 691 ◽  
pp. 487-497 ◽  
Author(s):  
Rabah Mehaddi ◽  
Olivier Vauquelin ◽  
Fabien Candelier
Keyword(s):  
Theoretical Model ◽  
Asymptotic Analysis ◽  
Flow Rate ◽  
Vertical Velocity ◽  
Analytical Solutions ◽  
Boussinesq Approximation ◽  
Singular Case ◽  
Maximal Height ◽  
Analytical Integration ◽  
Analytical Expressions

AbstractThis paper theoretically investigates the initial up-flow of a vertical turbulent fountain (round or plane) in a linearly stratified environment. Conservation equations (volume, momentum and buoyancy) are written under the Boussinesq approximation assuming an entrainment proportional to the vertical velocity of the fountain. Analytical integration leads to exact values of both density and flow rate at the maximal height reached by the fountain. This maximal height is expressed as a function of the release conditions and the stratification strength and plotted from a numerical integration in order to exhibit overall behaviour. Then, analytical expressions for the maximal height are derived from asymptotic analysis and compared to experimental correlations available for forced fountains. For weak fountains, these analytical expressions constitute a new theoretical model. Finally, modified expressions are also proposed in the singular case of an initially non-buoyant vertical release.

scholarly journals Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

2003 ◽  
Vol 2003 (2) ◽  
pp. 87-114 ◽  
Author(s):  
J. R. Fernández ◽  
M. Sofonea
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Mechanical Damage ◽  
Parabolic Type ◽  
Approximate Solutions ◽  
Damage Function ◽  
Optimal Order ◽  
Order Error ◽  
Fully Discrete ◽  
Solution Regularity

We consider the quasistatic Signorini′s contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

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